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Let $X$ be a complex toric variety equipped with the action of an algebraictorus $T$, and let $G$ be a complex linear algebraic group. We classify all$T$-equivariant principal $G$-bundles $\mathcal{E}$ over $X$ and the morphismsbetween them. When $G$ is connected and reductive, we characterize theequivariant automorphism group $\text{Aut}_T(\mathcal{E} )$ of $\mathcal{E}$ asthe intersection of certain parabolic subgroups of $G$ that arise naturallyfrom the $T$-action on $\mathcal{E}$. We then give a criterion for theequivariant reduction of the structure group of $\mathcal{E}$ to a Levisubgroup of $G$ in terms of $\text{Aut}_T(\mathcal{E} )$. We use it to prove aprincipal bundle analogue of Kaneyama's theorem on equivariant splitting oftorus equivariant vector bundles of small rank over a projective space. When$X$ is projective and $G$ is connected and reductive, we show that the notionsof stability and equivariant stability are equivalent for any $T$-equivariantprincipal $G$-bundle over $X$.

Classification, reduction and stability of toric principal bundles